%I #6 Jan 12 2015 17:17:22

%S 1,-3,2,-3,-22,3,-3,122,-69,4,-3,-518,891,-156,5,-3,1882,-8709,3444,

%T -295,6,-3,-6182,71931,-57036,9785,-498,7,-3,18906,-530181,789684,

%U -241095,23022,-777,8,-3,-54822,3598587,-9661260,4919865,-783378,47607,-1144,9,-3,152538,-22943493,107911860,-87977415,21896622,-2129673,89576,-1611,10

%N Triangle read by rows: T(n,k) appears in the transformation Sum_{k=0..n} (k+1)*x^k = Sum_{k=0..n} T(n,k)*(x+2k)^k.

%C Consider the transformation 1 + 2x + 3x^2 + 4x^3 + ... + (n+1)*x^n = T(n,0)*(x+0)^0 + T(n,1)*(x+2)^1 + T(n,2)*(x+4)^2 + ... + T(n,n)*(x+2n)^n for n >= 0.

%F T(n,n) = n+1, for n >= 0.

%F T(n,n-1) = n*(1 - 2*n - 2*n^2), for n >= 1.

%F T(n,n-2) = (n-1)*(2*n^4-2*n^3-6*n^2+2*n+1), for n >= 2.

%F T(n,n-3) = (2-n)*(4*n^6-24*n^5+26*n^4+54*n^3-72*n^2+9)/3, for n >= 3.

%e From - _Wolfdieter Lang_, Jan 12 2015: (Start)

%e The triangle T(n,k) starts:

%e n\k 0 1 2 3 4 5 6 7 8 9 ...

%e 0: 1

%e 1: -3 2

%e 2: -3 -22 3

%e 3: -3 122 -69 4

%e 4: -3 -518 891 -156 5

%e 5: -3 1882 -8709 3444 -295 6

%e 6: -3 -6182 71931 -57036 9785 -498 7

%e 7: -3 18906 -530181 789684 -241095 23022 -777 8

%e 8: -3 -54822 3598587 -9661260 4919865 -783378 47607 -1144 9

%e 9 : -3 152538 -22943493 107911860 -87977415 21896622 -2129673 89576 -1611 10

%e ... Reformatted.

%e ----------------------------------------------------------------------------------

%e n = 3: 1 + 2*x + 3*x^2 + 4*x^3 = -3*(x+0)^0 + 122*(x+2)^1 - 69*(x+4)^2 + 4* (x+6)^3. (End)

%o (PARI) T(n,k) = (k+1)-sum(i=k+1,n,(2*i)^(i-k)*binomial(i,k)*T(n,i))

%o for(n=0,10,for(k=0,n,print1(T(n,k),", ")))

%Y Cf. A247236.

%K sign,tabl

%O 0,2

%A _Derek Orr_, Dec 30 2014

%E Edited; - _Wolfdieter Lang_, Jan 12 2015